Please read carefully through the arguments and counterarguments.

Interpretation 1: They were drunk

Summary:  They brought their sacrifice while they were under the influence of strong drink (according to rabbinical tradition).

Arguments in favour of this view:

For this reason, immediately after this incident (Leviticus 10:8–9) the LORD forbids Aaron and his descendants to use strong drink. They are to abstain from strong drink in order to distinguish between the holy and the common (Leviticus 10:10).

Arguments against this view:

Leviticus 10:1 puts all the attention on the unauthorized (or strange) fire with which Nadab and Abihu brought their sacrifice. This was contrary to what God had commanded. Nothing is said here about a potential inebriation, a wrong motivation, or anything like it.

Interpretation 2: They sacrificed the incense together

Summary: Nadab and Abihu, serving as priests, wanted to sacrifice the incense together, at the same time. According to the regulations of the LORD, however, only one priest was allowed to do this.

Arguments in favour of this view:

1.  Repeatedly we read in God’s Word that one man brings the incense offering (see Exodus 30:7; Exodus 40:26–27; Leviticus 16:12–13; 1 Samuel 2:28; Luke 1:8–9).

2.  Because of this unlawful action, the fire with which the priests approached the altar of incense became unauthorized fire.

Arguments against this view:

1.  It is indeed possible that this, too, was against God’s precepts, but it is not mentioned in Leviticus 10:1 as the actual reason for God’s wrath. It was not their unauthorized simultaneous action that evoked God’s wrath, but the unauthorized fire with which they came to the altar of incense.

2.  If the simultaneous action of both priests was the reason God killed them, we might expect Leviticus 10:1 to say that this caused their entire sacrifice of incense to be unauthorized, and not only the fire they brought in their censers.

Interpretation 3: They did not have the right to take the fire from the altar

Summary: Nadab and Abihu, being ordinary priests, did not have the right to take fire from the altar in the courtyard of the sanctuary for their incense offerings.

Arguments in favour of this view:

Only the high priest was allowed to take fire from the altar in the courtyard for bringing (or having brought) the incense into the sanctuary.

Arguments against this view:

1.  Indeed, the igniting and keeping the incense offering burning is seen as the task of Aaron a few times in the OT (Exodus 30:7, Leviticus 16:12, and 1 Samuel 2:28). This might indicate that it was the high priest only who was permitted to perform this task.

2.  However, from Exodus 28:40 and onward the priestly service of Aaron and his sons is always mentioned. When in Exodus 30 only Aaron is mentioned in some activities, this does not mean that the task was not also something his sons could perform. It is more plausible that Aaron is designated here as the primary person who carries the responsibility, serving as the supervisor, while his sons might perform this as well.

3.  In Deuteronomy 33:8–10 the offering of the incense is also described more broadly as the task of Levi’s posterity.

4.  From the duties that the high priest had to perform especially on the Day of Atonement (Leviticus 16:1–34) it should not be inferred without question that he was also the only person allowed to perform those duties in the regular daily sacrificial service.

5.  From Luke 1:8–9 it appears that Zacharias, an ordinary priest, was appointed by lot to bring the incense offering.

6.  It is possible that, according to God’s decree, the high priest always had to transfer the fire from the bronze altar into a censer outside, after which one of the ordinary priests would add the incense and bring it to the altar of incense. This may also have been the case with the priest Zechariah in Luke 1:8–9. But if it were the error of Nadab and Abihu that they did not respect the special position of the high priest, wouldn’t this have been stated more clearly in verse 1? Here, however, the focus is not on a wrong action of the person(s) performing the sacrifice, but on the wrong fire they brought. 

Interpretation 4: They prepared the incense wrongly

Summary: Nadab and Abihu did not carefully prepare the incense according to the precept given in Exodus 30:34–38.

Arguments in favour of this view:

The LORD already warned in Exodus 30:9 that no strange (same word used as in Leviticus 10:1!) incense should be offered on the altar of incense.

Arguments against this view:

It does not say in Leviticus 10:1 that Nadab and Abihu brought a wrong incense, but that they brought a wrong fire before the LORD.

Interpretation 5: They did not use the fire from the altar of burnt offerings

Summary: Nadab and Abihu did not come with fire from the altar of burnt offerings and sin offerings into the courtyard for the incense offering in the sanctuary.

Arguments in favour of this view:

1.  In the description of Leviticus 10:1, the emphasis is on the strange fire that was brought before God.

2.  In Leviticus 16:12–14 we read that the fire for the incense offering was to be taken from the altar of burnt offering and sin offering after a sin offering had first been brought on it. There was to be no sacrifice of incense without a sin offering first. The fire linked the two altars. This applies also to the blood of the sacrificial animal: a portion of it had to be sprinkled by the priest on the horn of the altar of incense (see Leviticus 4:3–7).

3.  The incense offering was a symbolic representation of the prayers with which the priest and the people came before God to implore his mercy (see Luke 1:8–10). By linking this sacrifice to the sin offering the LORD wanted to teach his people that this petition for grace could only be granted because of the atonement of sins (cf. Heidelberg Catechism, Lord’s Day 45, q/a 117: the second and the third aspects belonging to prayer).

4.  Already at the dedication of the tabernacle, Aaron’s sons broke this indispensable link between the altar for the sin offering and the altar for the incense offering, acting contrary to God’s command. That is why the LORD felt compelled to punish this error as severely as he did.